∫secxdx
To solve the integral of sec(x) with respect to x, we can use the technique of integration by substitution
To solve the integral of sec(x) with respect to x, we can use the technique of integration by substitution. Let’s go step-by-step:
1. First, recall the derivative of the secant function, which is sec(x)tan(x). This will help us identify the substitution to use.
2. Let’s choose u = sec(x) + tan(x). To find du, we need to differentiate u with respect to x. The derivative of u can be obtained using the chain rule.
du/dx = d/dx(sec(x) + tan(x))
= sec(x)tan(x) + sec^2(x)
= sec(x)(sec(x) + tan(x))
= sec^2(x) + sec(x)tan(x)
3. Rearranging the equation du/dx = sec^2(x) + sec(x)tan(x), we get du = sec^2(x) + sec(x)tan(x) dx.
4. Now, substitute u and du into the integral:
∫sec(x)dx = ∫1 * sec(x)dx
= ∫1 * ((sec^2(x) + sec(x)tan(x))/(sec^2(x) + sec(x)tan(x))) dx
5. We now have the integral in terms of u:
∫sec(x)dx = ∫(sec^2(x) + sec(x)tan(x))/(sec^2(x) + sec(x)tan(x)) dx
= ∫du/(sec^2(x) + sec(x)tan(x))
6. Notice that the denominator sec^2(x) + sec(x)tan(x) is equivalent to u. Therefore, we can simplify the integral further:
∫du/u = ln|u| + C
7. However, we need to express the solution in terms of x. Recall that we set u = sec(x) + tan(x). Hence, we substitute back:
∫sec(x)dx = ln|sec(x) + tan(x)| + C
Therefore, the integral of sec(x) with respect to x is ln|sec(x) + tan(x)| + C, where C is the constant of integration.
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